# Lesson 4 Annuities: The Mathematics of Regular Payments

Size: px
Start display at page:

Transcription

1 Lesson 4 Annuities: The Mathematics of Regular Payments Introduction An annuity is a sequence of equal, periodic payments where each payment receives compound interest. One example of an annuity is a Christmas club at a bank. Here you make deposits each week or month to build up a Christmas fund. Another example is the payments you make on a car loan. In order to understand the behavior of annuities we need to investigate the mathematical idea of a sequence and, in particular, a geometric sequence. Geometric Sequences A sequence is a collection of numbers in a particular order. The individual numbers are called the terms of the sequence. Example 1 1, 5, 9, 13, Can you see how the terms were generated? Example 2 3, 1, 1/3, 1/9, Can you see how the terms were generated? Example 3 1, 1, 2, 3, 5, 8, 13, Can you see how the terms were generated? This is a famous sequence called the Fibonacci sequence. In our study of annuities we will need a particular type of sequence called a geometric sequence. A geometric sequence starts with an initial term P and from then on every term in the sequence is obtained by multiplying the preceding term in the sequence by the same constant c. The number c is called the common ratio of the geometric sequence. (We will see why in a minute.) Here are some examples. Example 4 5, 10, 20, 40, 80, Initial term = Note: case), a common ratio. Common ratio =. The ratio of any term to the preceding one is constant (2 in this Example 5 27, 9, 3, 1, 1/3, Initial term = Common ratio = Example 6 27, -9, 3, -1, 1/3, Initial term = Common ratio =

2 Notation: A geometric sequence with initial term P and common ratio c can be written as P, cp, c 2 P, c 3 P, c 4 P, We will label the terms using a common letter in this case G for geometric. We will use subscripts to indicate which term of the sequence we are referring to. We will start with 0 rather than 1 so that the subscript matches the power of c in the term. G 0 = P, G 1 = c 1 P, G 2 = c 2 P, G 3 = c 3 P, Summary For a geometric sequence G 0, G 1, G 2, we have 1) G 0 = P and G N = c G N 1 (called a recursive formula) 2) G N = c N P (called an explicit formula) Geometric sequences play an important role in the world of finance. Consider the following example. Example 7 Consider a geometric sequence with initial term P = 5000 and common ration c = Then the first few terms of this sequence are : G 0 = 5000 G 1 = (1.06)5000 = 5300 G 2 = (1.06) = 5618 G 3 = (1.06) = , etc. Notice that if we put dollar signs in front of these numbers, we get the principal and the future values over the first 3 years of an investment with a principal of \$5000 and an interest rate of 6% compounded annually! Let s see if we can generalize this example. Suppose that we have a principal P and an interest rate per period of i. Then the balances in the account at the end of each compounding period are the terms of a geometric sequence with initial term P and common ratio (1 + i)! P, P(1 + i), P(1 + i) 2, P(1 + i) 3, In discussing how much money will accumulate in an annuity (like a Christmas club) we will need to add up all the terms in such a geometric sequence (and there may be lots of them!). Thus, it would be great if we had a simple way to add up all the terms in a geometric sequence. Geometric Sequence Sum Formula Problem: Find the sum P + cp + c 2 P + c 3 P + + c N 1 P. To find the sum we are going to use a clever strategy that will eliminate most of the terms that we need to add up! Step 1 Multiply each term in given sum above by c: c(p + cp + c 2 P + c 3 P + + c N 1 P) = cp + c 2 P + c 3 P + + c N P

3 Step 2 Take the result from Step 1 and then subtract the original sum from it. (cp + c 2 P + c 3 P + + c N 1 P + c N P) ( P + cp + c 2 P + c 3 P + + c N 1 P) = c N P P = P(c N 1) Notice that almost all the terms have canceled out, appearing once with a plus sign and once with a minus sign! If we denote the original sum that we wanted to find by S (for sum!), then in Step 1 we constructed cs while is Step 2 we found cs S = S(c 1). Thus we have S(c 1) = P(c N 1) or, solving for S we get: S = or P + cp + c 2 P + c 3 P + + c N 1 P =. Notes: 1. The formula fails for c = 1. Why? (What is the sum of first N terms if c = 1?) 2. If you think of the left hand side as the sum of the first N terms of a geometric sequence and then think of the exponent of the right hand side as being one more than the highest power of c on the left hand side (or the number of terms being added ), then you have a simple way to remember the formula.

4 Future Value of an Annuity Let s begin by looking at examples of the 2 most common types of annuity: ordinary annuity and annuity due. Example 1. On August 10 Sherah joins a Christmas club at her bank. She will make \$200 deposits on the first of each of the next 3 months and on December 1 will be able to withdraw her money for shopping. Assume that the interest rate will be 6% compounded monthly. How much will be in her account on December 1? Solution: To do this problem we have to track the future value of each payment separately and then combine the results. The September 1 deposit will earn interest for 3 periods (months) where the rate per period is so FV Sept = 200(1.005) 3 = \$ The October 1 deposit will earn interest for 2 periods (months) so FV Oct = 200(1.005) 2 = \$ The November 1 deposit will earn interest for only 1 period so FV Nov = 200(1.005) 1 = \$ Thus, the future value of the annuity is given by the sum of these 3 future values: FV = \$ \$ \$201 = \$ Sherah earned \$6.03 in interest. Before we do our next example, let s look at some vocabulary associated with annuities. 1) The payment period of an annuity is the time between payments. 2) The term of an annuity is the time from the beginning of the first payment period to the end of the last payment period. (Sherah s annuity had a term of 3 months.) 3) An annuity is said to have expired at the end of its term. 4) An annuity is called simple if its compounding period is the same as its payment period. All of our annuities will be simple annuities. 5) An annuity due is an annuity for which each payment is due at the beginning of each payment period. 6) An ordinary annuity is an annuity for which each payment is due at the end of its payment period. Sherah s annuity is an example of an annuity due. The next example is an ordinary annuity. Example 2. Dan joins a Christmas club for September, October and November with interest at 6% compounded monthly. He makes \$200 payments at the end of each month. How much will be in his account for Christmas shopping on December 1? Solution: We again think of the future value of each payment separately and then combine the results.

5 We know that i =.005 again and so we have the following future values. FV Sept = 200(1.005) 2 = \$202.01, FV Oct = 200(1.005) 1 = \$ and FV Nov = 200(1.005) 0 or just \$200. The total for Dan is \$603.01, and he earned \$3.01 interest. Question: Why did Sherah earn more interest than Dan? Answer: Because each payment in her annuity earned interest for 1 more month. Let s compare the two annuities side-by-side to see exactly how they are related. Sept. Oct. Nov. FV Sherah = \$ = 200(1.005) (1.005) (1.005) 1 FV Dan = \$ = 200(1.005) (1.005) Notice that if we multiply each term in Dan s future value by 1.005, (1 + i), we will get Sherah s future value exactly; that is FV Dan (1.005) = FV Sherah. This shows us a general principle about the relationship between the future of an ordinary annuity and an annuity due, assuming that the interest rate and the terms are the same. In general, FV Ordinary (1 + i) = FV Due. We can convert an ordinary annuity to an annuity due by leaving all funds accumulated in the account for one additional period. If Dan leaves his \$ in his account for the month of December he will have (1.005) = \$ which is the same as Sherah s future value! Example 3. If you deposit \$100 every 6 months into an ordinary annuity paying 6% compounded semiannually, how much money would you have after a term of 3 years? Solution: Let s do a timeline and track the future value of each payment, adding them up at the end years periods deposits FV 1 = 100(1.03) 5 = \$115.93, FV 2 = 100(1.03) 4 = \$112.55, FV 3 = 100(1.03) 3 = \$109.27, FV 4 = 100(1.03) 2 = \$106.09, FV 5 = 100(1.03) 1 = \$ and FV 6 = \$100 for a total of \$ What would the future value have been if we had used an annuity due with same interest rate? Answer: (1.03) = \$

6 Since there were only 6 payments it was fairly easy to look at each future value separately and then add up the results. But what if there were 200 payments? It would be nice to have a fast, convenient way to add up the future values of all those payments, and, in fact, we already have the tools we need: the formula for the sum of a geometric sequence! We will denote our periodic payments by R and we will assume that our account is earning interest at a rate of i for a term of n periods (that is, there will be n payments). We construct the following time line n-3 n-2 n-1 n periods R R R R R R R R The last payment will earn no interest so its future value is just R. The next to last payment will earn interest for one period and so its future value will be R(1 + i) 1. The second to last payment will earn interest for two periods and so its future value will be R(1 + i) 2. We continue working backwards this way and we find that the second payment will earn interest for all but the first 2 periods, that is it will interest for n 2 periods and so will have a future value of R(1 + i) n-2. Similarly, the first payment will earn interest for n 1 periods (all but the first period) and will have a future value of R(1 + i) n-1. To find the future value F of the annuity, we now need to add up all the future values of the separate payments. F = R + R(1 + i) 1 + R(1 + i) 2 + R(1 + i) R(1 + i) n-2 + R(1 + i) n-1. Notice that this is a geometric sequence with first term R and with the common ratio c = 1 + i! Thus we can plug into our sum formula for a geometric sequence to find our future value formula. This is our formula for the future value of an ordinary annuity. For an annuity due, we need only multiply this result by 1 + i to get the correct future value. F due = F ordinary (1 + i) Let s look at some examples. Example 4 Verify the result of Example 3 by using the future value formula for an ordinary annuity. Solution: In Example 3 we had R = \$100, i =.03 (6% compounded semiannually) and n = 6 (semiannual payments for 3 years). Thus, F = Example 5 What is the future value of an ordinary annuity at the end of 5 years if \$100 per month is deposited in an account earning 9% compounded monthly? What would this future value be for an annuity due?

7 Solution: Here R = 100, n = kt = 12(5) = 60 periods, and i =.09/12 = Substituting into our formula for the future value of an ordinary annuity gives. For an annuity due, we only have to multiply the answer we just got by 1 + i = : (1.0075) = \$ Example 6 Jim puts \$200 per month into an ordinary annuity paying 8.75% compounded monthly for 35 years. Find the future value of his annuity. What would the future value be as an annuity due? Solution: Here R = 200, r =.0875, k = 12 and t = 35. Thus i =.0875/12 and n = 12(35) = 420 periods in 35 years. Using our formula,. For an annuity due, we simply multiply the future value for the ordinary annuity by 1 + i = /12: F Due = 552,539.96( /12) = \$556, Sinking Funds An account that is established to accumulate funds for a future obligation is called a sinking fund. If regular, periodic payments are made to this fund, we have an annuity. Example 7. A couple sets up a sinking fund for their new baby s education. How much should they have deducted from their bi-weekly (every two weeks) paycheck to have \$30,000 in 18 years at 9.25% interest? Assume an ordinary annuity. Solution: Here F = 30000, i = r/k =.0925/26 and n = kt = 26(18) = 468 total periods, and we want to find R, the periodic payment. Filling in the values in our formula for the future value of an ordinary annuity, we have. Solving for R gives R = or \$25 to the nearest penny. Example 8. A company estimates that it will need \$10,000 in 5 years to replace a piece of equipment. A sinking fund is established using an ordinary annuity with monthly payments earning interest at 6% compounded monthly. What should be payments be? Solution: We have F = 10,000, i = r/k =.06/12 =.005, and n = kt = 12(5) = 60. Substituting into our formula for the future value of an ordinary annuity gives Present Value of an Annuity. Solving for R gives R = \$ The present value of an annuity is the lump sum that can be deposited at the beginning of the annuity s term at the same interest rate and compounding period, and which would yield the same future value as

8 the annuity at the end of its term. We sometimes say that the lump sum will buy or generate the annuity. *We are looking for a present value P so that F compound = F annuity. Example 9. What lump sum would the company in Example 8 have to deposit now at 6% compounded monthly in order to have the same amount after 5 years as their annuity will give them? Solution: We know from Example 8 that their annuity will accumulate the \$10,000 they will need to replace their piece of equipment. So now we have a standard present value problem for compound interest. We know F = \$10,000, r =.06, k = 12 and t = 5. Thus i = r/k =.06/12 =.005, and n = kt = 60. Substituting these values in our formula gives = P(1.005) 60 or P = Example 10. Find the present value of an ordinary annuity that has \$200 monthly payments for 25 years if the account receives 10 ½ % interest (compounded monthly). Solution: To find the present value P, we need to solve the equation (*) for P. We have i =.105/12, n = 12(25) = 300 and R = \$200. Substituting and solving for P gives P = \$21, Note: We can use (*) to find a formula for P by dividing both sides by (1 + i) n : Example 11. What is the present value of an annuity that has \$200 monthly payments for 5 years with a rate of 6% (compounded monthly)? Solution: We know that R = \$200, i = r/k =.06/12 =.005 and n = kt = 12(5) = 60. We will substitute into the formula above.

### 1. Annuity a sequence of payments, each made at equally spaced time intervals.

Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology

### Chapter 3 Mathematics of Finance

Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds Learning Objectives for Section 3.3 Future Value of an Annuity; Sinking Funds The student will be able to compute the

### 2. How would (a) a decrease in the interest rate or (b) an increase in the holding period of a deposit affect its future value? Why?

CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equal-sized

### Chapter 6. Time Value of Money Concepts. Simple Interest 6-1. Interest amount = P i n. Assume you invest \$1,000 at 6% simple interest for 3 years.

6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in

### Finance Notes ANNUITIES

Annuities Page 1 of 8 ANNUITIES Objectives: After completing this section, you should be able to do the following: Calculate the future value of an ordinary annuity. Calculate the amount of interest earned

### first complete "prior knowlegde" -- to refresh knowledge of Simple and Compound Interest.

ORDINARY SIMPLE ANNUITIES first complete "prior knowlegde" -- to refresh knowledge of Simple and Compound Interest. LESSON OBJECTIVES: students will learn how to determine the Accumulated Value of Regular

### Annuities and Sinking Funds

Annuities and Sinking Funds Sinking Fund A sinking fund is an account earning compound interest into which you make periodic deposits. Suppose that the account has an annual interest rate of compounded

### Chapter The Time Value of Money

Chapter The Time Value of Money PPT 9-2 Chapter 9 - Outline Time Value of Money Future Value and Present Value Annuities Time-Value-of-Money Formulas Adjusting for Non-Annual Compounding Compound Interest

### Finance 197. Simple One-time Interest

Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for

### Chapter 6. Discounted Cash Flow Valuation. Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Answer 6.1

Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation

### CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: \$5,000.08 = \$400 So after 10 years you will have: \$400 10 = \$4,000 in interest. The total balance will be

### Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3)

MATH 1003 Calculus and Linear Algebra (Lecture 3) Future Value of an Annuity Definition An annuity is a sequence of equal periodic payments. We call it an ordinary annuity if the payments are made at the

### Stats & Discrete Math Annuities Notes

Stats & Discrete Math Annuities Notes Many people have long term financial goals and limited means with which to accomplish them. Your goal might be to save \$3,000 over the next four years for your college

### Check off these skills when you feel that you have mastered them.

Chapter Objectives Check off these skills when you feel that you have mastered them. Know the basic loan terms principal and interest. Be able to solve the simple interest formula to find the amount of

### Finding the Payment \$20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = \$488.26

Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive \$5,000 per month in retirement.

### Mathematics. Rosella Castellano. Rome, University of Tor Vergata

and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings

### PowerPoint. to accompany. Chapter 5. Interest Rates

PowerPoint to accompany Chapter 5 Interest Rates 5.1 Interest Rate Quotes and Adjustments To understand interest rates, it s important to think of interest rates as a price the price of using money. When

### MAT116 Project 2 Chapters 8 & 9

MAT116 Project 2 Chapters 8 & 9 1 8-1: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the

### CHAPTER 1. Compound Interest

CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.

### Chapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams

Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present

### Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

### Chapter F: Finance. Section F.1-F.4

Chapter F: Finance Section F.1-F.4 F.1 Simple Interest Suppose a sum of money P, called the principal or present value, is invested for t years at an annual simple interest rate of r, where r is given

### Discounted Cash Flow Valuation

6 Formulas Discounted Cash Flow Valuation McGraw-Hill/Irwin Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline Future and Present Values of Multiple Cash Flows Valuing

### CHAPTER 6 DISCOUNTED CASH FLOW VALUATION

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and

### Finance. Simple Interest Formula: I = P rt where I is the interest, P is the principal, r is the rate, and t is the time in years.

MAT 142 College Mathematics Finance Module #FM Terri L. Miller & Elizabeth E. K. Jones revised December 16, 2010 1. Simple Interest Interest is the money earned profit) on a savings account or investment.

### Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

### Finance CHAPTER OUTLINE. 5.1 Interest 5.2 Compound Interest 5.3 Annuities; Sinking Funds 5.4 Present Value of an Annuity; Amortization

CHAPTER 5 Finance OUTLINE Even though you re in college now, at some time, probably not too far in the future, you will be thinking of buying a house. And, unless you ve won the lottery, you will need

### TIME VALUE OF MONEY (TVM)

TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate

### Geometric Series and Annuities

Geometric Series and Annuities Our goal here is to calculate annuities. For example, how much money do you need to have saved for retirement so that you can withdraw a fixed amount of money each year for

### UNIT AUTHOR: Elizabeth Hume, Colonial Heights High School, Colonial Heights City Schools

Money & Finance I. UNIT OVERVIEW & PURPOSE: The purpose of this unit is for students to learn how savings accounts, annuities, loans, and credit cards work. All students need a basic understanding of how

### Finite Mathematics. CHAPTER 6 Finance. Helene Payne. 6.1. Interest. savings account. bond. mortgage loan. auto loan

Finite Mathematics Helene Payne CHAPTER 6 Finance 6.1. Interest savings account bond mortgage loan auto loan Lender Borrower Interest: Fee charged by the lender to the borrower. Principal or Present Value:

### Chapter 03 - Basic Annuities

3-1 Chapter 03 - Basic Annuities Section 7.0 - Sum of a Geometric Sequence The form for the sum of a geometric sequence is: Sum(n) a + ar + ar 2 + ar 3 + + ar n 1 Here a = (the first term) n = (the number

### How To Calculate A Pension

Interests on Transactions Chapter 10 13 PV & FV of Annuities PV & FV of Annuities An annuity is a series of equal regular payment amounts made for a fixed number of periods 2 Problem An engineer deposits

### CHAPTER 17 ENGINEERING COST ANALYSIS

CHAPTER 17 ENGINEERING COST ANALYSIS Charles V. Higbee Geo-Heat Center Klamath Falls, OR 97601 17.1 INTRODUCTION In the early 1970s, life cycle costing (LCC) was adopted by the federal government. LCC

### Compound Interest Formula

Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt \$100 At

### Calculations for Time Value of Money

KEATMX01_p001-008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with

### 5 More on Annuities and Loans

5 More on Annuities and Loans 5.1 Introduction This section introduces Annuities. Much of the mathematics of annuities is similar to that of loans. Indeed, we will see that a loan and an annuity are just

### The Time Value of Money C H A P T E R N I N E

The Time Value of Money C H A P T E R N I N E Figure 9-1 Relationship of present value and future value PPT 9-1 \$1,000 present value \$ 10% interest \$1,464.10 future value 0 1 2 3 4 Number of periods Figure

### The Compound Amount : If P dollars are deposited for n compounding periods at a rate of interest i per period, the compound amount A is

The Compound Amount : If P dollars are deposited for n compounding periods at a rate of interest i per period, the compound amount A is A P 1 i n Example 1: Suppose \$1000 is deposited for 6 years in an

### Finance Unit 8. Success Criteria. 1 U n i t 8 11U Date: Name: Tentative TEST date

1 U n i t 8 11U Date: Name: Finance Unit 8 Tentative TEST date Big idea/learning Goals In this unit you will study the applications of linear and exponential relations within financing. You will understand

### Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations

Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the Hewlett-Packard

### 5. Time value of money

1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

### Sample problems from Chapter 10.1

Sample problems from Chapter 10.1 This is the annuities sinking funds formula. This formula is used in most cases for annuities. The payments for this formula are made at the end of a period. Your book

### Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material. i = 0.75 1 for six months.

Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material 1. a) Let P be the recommended retail price of the toy. Then the retailer may purchase the toy at

### 5.1 Simple and Compound Interest

5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

### Lesson 1. Key Financial Concepts INTRODUCTION

Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have

### Dick Schwanke Finite Math 111 Harford Community College Fall 2015

Using Technology to Assist in Financial Calculations Calculators: TI-83 and HP-12C Software: Microsoft Excel 2007/2010 Session #4 of Finite Mathematics 1 TI-83 / 84 Graphing Calculator Section 5.5 of textbook

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Introduction to Real Estate Investment Appraisal

Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has

### Warm-up: Compound vs. Annuity!

Warm-up: Compound vs. Annuity! 1) How much will you have after 5 years if you deposit \$500 twice a year into an account yielding 3% compounded semiannually? 2) How much money is in the bank after 3 years

### CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount

### Annuities: Present Value

8.5 nnuities: Present Value GOL Determine the present value of an annuity earning compound interest. INVESTIGTE the Math Kew wants to invest some money at 5.5%/a compounded annually. He would like the

### TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY

TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction In this assignment we will discuss how to calculate the Present Value

### Problem Set: Annuities and Perpetuities (Solutions Below)

Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save \$300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years

### DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS

Chapter 5 DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS The basic PV and FV techniques can be extended to handle any number of cash flows. PV with multiple cash flows: Suppose you need \$500 one

### Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Chapter Outline. Multiple Cash Flows Example 2 Continued

6 Calculators Discounted Cash Flow Valuation Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

### 300 Chapter 5 Finance

300 Chapter 5 Finance 17. House Mortgage A couple wish to purchase a house for \$200,000 with a down payment of \$40,000. They can amortize the balance either at 8% for 20 years or at 9% for 25 years. Which

### How to calculate present values

How to calculate present values Back to the future Chapter 3 Discounted Cash Flow Analysis (Time Value of Money) Discounted Cash Flow (DCF) analysis is the foundation of valuation in corporate finance

### CHAPTER 4 DISCOUNTED CASH FLOW VALUATION

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. Assuming positive cash flows and interest rates, the future value increases and the present value

### Compounding Quarterly, Monthly, and Daily

126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,

### Time Value Conepts & Applications. Prof. Raad Jassim

Time Value Conepts & Applications Prof. Raad Jassim Chapter Outline Introduction to Valuation: The Time Value of Money 1 2 3 4 5 6 7 8 Future Value and Compounding Present Value and Discounting More on

### 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%?

Chapter 2 - Sample Problems 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%? 2. What will \$247,000 grow to be in

### The Time Value of Money

C H A P T E R6 The Time Value of Money When plumbers or carpenters tackle a job, they begin by opening their toolboxes, which hold a variety of specialized tools to help them perform their jobs. The financial

### E INV 1 AM 11 Name: INTEREST. There are two types of Interest : and. The formula is. I is. P is. r is. t is

E INV 1 AM 11 Name: INTEREST There are two types of Interest : and. SIMPLE INTEREST The formula is I is P is r is t is NOTE: For 8% use r =, for 12% use r =, for 2.5% use r = NOTE: For 6 months use t =

### Simple Interest. and Simple Discount

CHAPTER 1 Simple Interest and Simple Discount Learning Objectives Money is invested or borrowed in thousands of transactions every day. When an investment is cashed in or when borrowed money is repaid,

### CALCULATOR HINTS ANNUITIES

CALCULATOR HINTS ANNUITIES CALCULATING ANNUITIES WITH THE FINANCE APP: Select APPS and then press ENTER to open the Finance application. SELECT 1: TVM Solver The TVM Solver displays the time-value-of-money

### PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time.

PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values

### Module 5: Interest concepts of future and present value

Page 1 of 23 Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present and future values, as well as ordinary annuities

### Present Value Concepts

Present Value Concepts Present value concepts are widely used by accountants in the preparation of financial statements. In fact, under International Financial Reporting Standards (IFRS), these concepts

### Ing. Tomáš Rábek, PhD Department of finance

Ing. Tomáš Rábek, PhD Department of finance For financial managers to have a clear understanding of the time value of money and its impact on stock prices. These concepts are discussed in this lesson,

### Time Value of Money Concepts

BASIC ANNUITIES There are many accounting transactions that require the payment of a specific amount each period. A payment for a auto loan or a mortgage payment are examples of this type of transaction.

### Six Functions of a Dollar. Made Easy! Business Statistics AJ Nelson 8/27/2011 1

Six Functions of a Dollar Made Easy! Business Statistics AJ Nelson 8/27/2011 1 Six Functions of a Dollar Here's a list. Simple Interest Future Value using Compound Interest Present Value Future Value of

### The Institute of Chartered Accountants of India

CHAPTER 4 SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY APPLICATIONS SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY- APPLICATIONS LEARNING OBJECTIVES After studying this chapter students will be able

### Time Value of Money. Background

Time Value of Money (Text reference: Chapter 4) Topics Background One period case - single cash flow Multi-period case - single cash flow Multi-period case - compounding periods Multi-period case - multiple

### Financial Literacy in Grade 11 Mathematics Understanding Annuities

Grade 11 Mathematics Functions (MCR3U) Connections to Financial Literacy Students are building their understanding of financial literacy by solving problems related to annuities. Students set up a hypothetical

### Ordinary Annuities Chapter 10

Ordinary Annuities Chapter 10 Learning Objectives After completing this chapter, you will be able to: > Define and distinguish between ordinary simple annuities and ordinary general annuities. > Calculate

### 8.1 Simple Interest and 8.2 Compound Interest

8.1 Simple Interest and 8.2 Compound Interest When you open a bank account or invest money in a bank or financial institution the bank/financial institution pays you interest for the use of your money.

Chapter 13 Annuities and Sinking Funds McGraw-Hill/Irwin Copyright 2011 by the McGraw-Hill Companies, Inc. All rights reserved. #13 LU13.1 Annuities and Sinking Funds Learning Unit Objectives Annuities:

### FIN 3000. Chapter 6. Annuities. Liuren Wu

FIN 3000 Chapter 6 Annuities Liuren Wu Overview 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams Learning objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate

### Important Financial Concepts

Part 2 Important Financial Concepts Chapter 4 Time Value of Money Chapter 5 Risk and Return Chapter 6 Interest Rates and Bond Valuation Chapter 7 Stock Valuation 130 LG1 LG2 LG3 LG4 LG5 LG6 Chapter 4 Time

### Student Loans. The Math of Student Loans. Because of student loan debt 11/13/2014

Student Loans The Math of Student Loans Alice Seneres Rutgers University seneres@rci.rutgers.edu 1 71% of students take out student loans for their undergraduate degree A typical student in the class of

### Discounted Cash Flow Valuation

Discounted Cash Flow Valuation Chapter 5 Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

### Activity 3.1 Annuities & Installment Payments

Activity 3.1 Annuities & Installment Payments A Tale of Twins Amy and Amanda are identical twins at least in their external appearance. They have very different investment plans to provide for their retirement.

### The Time Value of Money

CHAPTER 7 The Time Value of Money After studying this chapter, you should be able to: 1. Explain the concept of the time value of money. 2. Calculate the present value and future value of a stream of cash

### Level Annuities with Payments More Frequent than Each Interest Period

Level Annuities with Payments More Frequent than Each Interest Period 1 Examples 2 Annuity-immediate 3 Annuity-due Level Annuities with Payments More Frequent than Each Interest Period 1 Examples 2 Annuity-immediate

### International Financial Strategies Time Value of Money

International Financial Strategies 1 Future Value and Compounding Future value = cash value of the investment at some point in the future Investing for single period: FV. Future Value PV. Present Value

### TIME VALUE OF MONEY. In following we will introduce one of the most important and powerful concepts you will learn in your study of finance;

In following we will introduce one of the most important and powerful concepts you will learn in your study of finance; the time value of money. It is generally acknowledged that money has a time value.

### Discounted Cash Flow Valuation

BUAD 100x Foundations of Finance Discounted Cash Flow Valuation September 28, 2009 Review Introduction to corporate finance What is corporate finance? What is a corporation? What decision do managers make?

### Section 8.1. I. Percent per hundred

1 Section 8.1 I. Percent per hundred a. Fractions to Percents: 1. Write the fraction as an improper fraction 2. Divide the numerator by the denominator 3. Multiply by 100 (Move the decimal two times Right)

### Place in College Curriculum: This course is required for all Business Administration AAS degree and Administrative Assistant certificate.

Salem Community College Course Syllabus Section I Course Title: Business Mathematics Course Code: BUS106 Lecture Hours: 3 Lab Hours: 0 Credits: 3 Course Description: The Business Mathematics course is

### 2.1 The Present Value of an Annuity

2.1 The Present Value of an Annuity One example of a fixed annuity is an agreement to pay someone a fixed amount x for N periods (commonly months or years), e.g. a fixed pension It is assumed that the

### Appendix C- 1. Time Value of Money. Appendix C- 2. Financial Accounting, Fifth Edition

C- 1 Time Value of Money C- 2 Financial Accounting, Fifth Edition Study Objectives 1. Distinguish between simple and compound interest. 2. Solve for future value of a single amount. 3. Solve for future

### A = P (1 + r / n) n t

Finance Formulas for College Algebra (LCU - Fall 2013) ---------------------------------------------------------------------------------------------------------------------------------- Formula 1: Amount

### MATHEMATICS OF FINANCE AND INVESTMENT

MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics

### Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

### In Section 5.3, we ll modify the worksheet shown above. This will allow us to use Excel to calculate the different amounts in the annuity formula,

Excel has several built in functions for working with compound interest and annuities. To use these functions, we ll start with a standard Excel worksheet. This worksheet contains the variables used throughout